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Properties

Label 2-2592-9.5-c0-0-2
Degree $2$
Conductor $2592$
Sign $0.766 - 0.642i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)5-s + 1.41i·17-s + (0.499 + 0.866i)25-s + (1.22 − 0.707i)29-s − 2·37-s + (1.22 + 0.707i)41-s + (0.5 − 0.866i)49-s + 1.41i·53-s + (−1 − 1.73i)61-s + (−1.00 + 1.73i)85-s − 1.41i·89-s + (−1.22 + 0.707i)101-s + (1.22 + 0.707i)113-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)5-s + 1.41i·17-s + (0.499 + 0.866i)25-s + (1.22 − 0.707i)29-s − 2·37-s + (1.22 + 0.707i)41-s + (0.5 − 0.866i)49-s + 1.41i·53-s + (−1 − 1.73i)61-s + (−1.00 + 1.73i)85-s − 1.41i·89-s + (−1.22 + 0.707i)101-s + (1.22 + 0.707i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.511161024\)
\(L(\frac12)\) \(\approx\) \(1.511161024\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + 2T + T^{2} \)
41 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.196128981026488736756168263800, −8.458414039422608606387242879769, −7.61081141874229527616605385560, −6.60884407378915453942880809018, −6.18637149471754256473959326497, −5.44438568912478778802267639814, −4.40813699877038880402756553684, −3.36047349439721701979502878004, −2.41739117741447100546893534561, −1.55201916167757218956749731574, 1.10229454457245794434333607890, 2.19333135351944996145448138200, 3.11499848502491450285321702187, 4.41951919850796388613845190883, 5.19037765228628931562854988146, 5.71103468131900396614555445444, 6.69588696644905937712662935316, 7.34366904173380135476371145570, 8.467233093468496509780913394705, 9.034026563243460805606484985757

Graph of the $Z$-function along the critical line